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International Journal of Advances in Science and Technology, Vol. 4, No 4, 2012 Cototal Domination in Semitotal- Point Graph B.Basavanagoud*, Prashant. V. Patil and Sunilkumar. M. Hosamani Department of Mathematics, Karnatak University, Dharwad-580 003,Karnataka, India Email*: b.basavanagoud@gmail.com Abstract Let be a graph. Then the semitotal-point graph is denoted by . Let the vertices and edges of be called the elements of . A dominating set of a graph is a cototal dominating set of if has no isolated vertices. The cototal domination number of denoted by of is the minimum cardinality of a cototal dominating set of . In this paper we study the cototal domination in semitotal-point graphs and obtain many bounds of in terms of elements of but not the elements of . Also its relationship with other domination parameters are established. 2000 Mathematics Subject Classification: 05C69. Keywords: domination number, cototal domination number. 1 Introduction. Throughout this paper by the word graph we mean a finite, nontrivial, undirected, connected, without loops or multiple edges. Let the vertices and edges of a graph be called the elements of . For notations and terminology we follow Harary [4]. The neighborhood of a vertex in is the set consisting of all vertices which are adjacent with . A set is a dominating set, if every vertex in is adjacent to some vertex in . The domination number of , is the minimum cardinality of a dominating set. A dominating set of a graph is a cototal dominating set of if every vertex is not an isolated vertex in the induced subgraph . The cototal domination number of , is the minimum cardinality of a cototal dominating set [7]. For any graph , semitotal-point graph is the graph whose vertex set is the union of vertices and edges in which two vertices are adjacent if and only if they are adjacent vertices of or one is a vertex and other is an edge of incident with it [8]. We now define the cototal domination number of the semitotal-point graph as follows: A dominating set of a graph is a cototal dominating set of if has no isolated vertices. The cototal domination number of denoted by of is the minimum cardinality of a cototal dominating set of . The following figure shows the formation of and . Special Issue Page 109 of 114 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 4, No 4, 2012 1 1 a a G: T2(G): 1 d 2 2 b d b 4 3 4 c 3 c Figure 1 In , , therefore . In , , , and so on. Therefore . 2 Main Results In the following theorem, we state the exact values of cototal domination number of of some standard class of graphs. Theorem 2.1 For any path ; , . For any cycle ; , . For any complete bipartite graph ; , . For any complete graph ; , For any wheel ; , . The following upper bound of is immediate from the above Theorem 2.1. Theorem 2.2 For any graph , . Further the equality holds for for and is odd. Now, we establish another upper bound for when is a tree. Theorem 2.3 For any tree , Proof. Let be a dominating set of and be a cutvertex in . Let be the corresponding edge vertices of in . Then, every may or may not be an isolated vertex in . Now, it follows that, ; Special Issue Page 110 of 114 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 4, No 4, 2012 form a cototal dominating set of . Hence . The next theorem gives an upper bound of in terms of order and minimum degree of . Theorem 2.4 If , then , where is the minimum degree of . Proof. Let be a vertex of minimum degree such that . Let be the set of edge vertices in with respective the edges of . Suppose . Then each edge vertex forms a cycle in and by definition of , has vertices and edges. Therefore, . Hence . In a graph if , then is called a pendant vertex of . Theorem 2.5 For any tree , , where is the number of vertices adjacent to pendant vertices. Proof. Suppose is a tree with vertices and be the set of vertices adjacent to pendant vertices of with size . Let be the set of edge vertices in with respective to the edges of . Since each edge vertex ; is adjacent to exactly two vertices and forms a cycle of length three in . Then, contains at least one isolated vertex. Let be the set of such vertices. Since and also and is the set of all pendant vertices of , therefore forms a cototal dominating set of . Hence number of pendant vertices of . Theorem A [3]. If is a graph without isolated vertices of order , then . Theorem 2.6 If contains a pendant vertex, then . Proof. Let be a vertex of degree one and the set be the set of edge vertices in with respective the edges of . Now we consider the following cases: Case 1. If and is a tree then by Theorem 2.3, . Case 2. If and is not a tree. Let and starting from , consider the alternate vertices of and that form a dominating set of . Since each vertex of belongs to at least one edge vertex in , therefore is a cototal dominating set of . Since is also a dominating set of , therefore by Theorem A, . Theorem 2.7 If is a tree in which the number of pendant vertices is same as the number of cutvertices and every cutvertex is adjacent to a pendant vertex, then , where is the number of pendant vertices or number of cutvertices of . Special Issue Page 111 of 114 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 4, No 4, 2012 Proof. Let be a tree with the above stated property. Then . Let be the cototal dominating set of . If set are the edges of , then the set will be corresponding edge vertices of . Since each ; is , then contains a cycle of length three and every ; is adjacent the set of all such vertices. Then has no isolated vertex. On the other hand, let , then the set forms a cototal dominating set of and , which is not possible. So . Now, it follows that every has at least one neighbor in and . Thus the result. Theorem B [5]. For any connected graph of order , where is the maximum degree of . Next we obtain a lower bound of in terms of order, size and maximum degree of . Theorem 2.8 For any graph , Proof. Since every cototal dominating set is a dominating set, therefore . Also by Theorem B, we get the required result. Theorem C [7]. For any graph , if and only if each component of is a star. Theorem 2.9 In a tree , if each component is a star, then . Proof. Let the set be the set of all center vertices of stars in a tree. Then, . But by Theorem C, . Hence from these two we get the result. Theorem 2.10 If is an independent dominating set of both and , then . Proof. Let be the set of independent vertices of which also a dominating set of and . Also by definition of , each edge vertex forms a and is independent, therefore has no isolated vertex. Therefore . A dominating set of a graph , is called a maximal dominating set, if is not a dominating set of . The maximal domination number of , is the minimum cardinality of a maximal dominating set [6 ]. Theorem D [6]. For any graph , exactly one of these holds: Special Issue Page 112 of 114 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 4, No 4, 2012 Theorem 2.11 For any graph , . Proof. Let be a maximal dominating set of and if for a vertex is an isolated vertex, then is a cototal dominating set of . If is not an isolated vertex, then itself forms a cototal dominating set of . Hence it follows from result of Theorem D, . Thus the result. Theorem 2.12 For any graph with , . Proof. Let be a connected graph. Let and be the cototal dominating sets of and respectively. Suppose forms a dominating set of . Let set be the set of edge vertices in with respect to the edges of . Then every ; is adjacent with exactly two point vertices of in . Since and has no isolated vertices, therefore it follows that forms a cototal dominating set of . Hence . Therefore . A dominating set is a total dominating set if the induced subgraph has no isolated vertex. The total domination number of a graph is the minimum cardinality of a total dominating set. The vertex independence number is the maximum cardinality among the independent set of vertices of . Theorem 2.13 For any graph , . Proof. Let be any graph without isolates and be a total dominating set of . Suppose the induced subgraph has no isolates then is a cototal dominating set of . If there exists a set such that each vertex in is an isolated vertex in . Thus forms a cototal dominating set of and . Thus by above Theorem 2.12, . Nordhaus-Gaddum type results. Theorem 2.14 Let a graph and its complement be connected. Then . Acknowledgement This research was supported by UGC-SAP DRS-II New Delhi, India: for 2010-2015. References [1] B.Basavanagoud, S.M.Hosamani and S.H.Malghan, Domination in semitotal-point graphs, J.Comp. and Math.Sci., Vol.1(5)(2010), 598-605. Special Issue Page 113 of 114 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 4, No 4, 2012 [2] E. J. Cockayne, R.M.Dawes and S. T. Hedetniemi, Total domination in graphs, Networks , 10(1980), 211-219. [3] E. J. Cockayne and S. T. Hedetniemi, Towards a theory of domination in Graphs, Networks , 7(1977), 247-261. [4] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, (1969). [5] T.W.Haynes, S.T.Hedetniemi and P.J.Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc, New York (1998). [6] V.R.Kulli and B.Janakiram, The maximal domination number of a graph, Graph Theory Notes of New York, New York Academy of Sciences, XXXIII(1997), 11-13. [7] V.R.Kulli, B.Janakiram and R.R.Iyer, The cototal domination number of a graph J.Discrete Mathematical Sciences and Cryptography, 2(1999), 179-184. [8] E.Sampathkumar and S.B.Chikkodimath, The semi-total graphs of a graph-I, Journal of the Karnatak University-Science, XVIII(1973), 274-280. Authors Profile Dr. B.Basavanagoud received his both MSc and Ph.D degree in Mathematics from Gulbarga University, Gulbarga, India. Currently he is working as Professor and Chairman, Department of Mathematics, Karnatak University, Dharwad. He has published many research papers in reputed national and international jourmnals. His area of interest lies mainly in Graph Theory. He is also life member of The Indian Mathematical Society, The Indian Science Congress Association and International Journal of Mathematical Sciences and Engineering Applications. He chaired the session in the International Congress of Mathematicians(ICM-2010) and delivered a contributed talk at the same Special Issue Page 114 of 114 ISSN 2229 5216